Bercovier-Engelman test

The exact solution of Bercovier-ENgelman [BE79] is defined on \(\Omega=\[0, 1]^3\) by:

\[\begin{eqnarray} u_1 &=& -256y(y-1)(2y-1)x^2(x-1)^2\\ u_2 &=& 256x(x-1)(2x-1)y^2(y-1)^2\\ u_3 &=& 0\\ p &=& (x-0.5)(y-0.5) \end{eqnarray}\]

External forces are calculated using exact solution in the Stokes / Navier-Stokes equations.

Dirichlet conditions

We compare the approximated solution and the exact solution of Bercovier-Engelman, respectively for pressure (Figure 1 and Figure 2) and velocity (Figure 3 and Figure 4).

Stokes BE P
Figure 1. Pressure on the unit cube. Computed solution
Stokes BE Pexact
Figure 2. Pressure on the unit cube. Exact solution
Stokes BE V
Figure 3. Velocity field on the unit cube on the cut z=0.5. Computed solution
Stokes BE Vexact
Figure 4. Velocity field on the unit cube on the cut z=0.5. Exact solution

FreeFem++ algorithm:

  • Stokes_BE_dirichlet.edp

  • Navier-Stokes_BE_dirichlet.edp

Mixed conditions

We compare the approximated solution and the exact solution of Bercovier-Engelman, respectively for pressure (Figure 5 and Figure 6) and velocity (Figure 7 and Figure 8).

Stokes BE P Mixed
Figure 5. Pressure on the unit cube. Computed solution
Stokes BE P Mixed
Figure 6. Pressure on the unit cube. Exact solution
Stokes BE V Mixed
Figure 7. Velocity field on the unit cube on the cut z=0.5. Computed solution
Stokes BE Vexact Mixed
Figure 8. Velocity field on the unit cube on the cut z=0.5. Exact solution